L(s) = 1 | + (0.604 − 1.62i)3-s − 0.533·5-s + (−2.26 − 1.96i)9-s + 3.92i·11-s + (−0.116 + 0.0674i)13-s + (−0.322 + 0.866i)15-s + (−2.16 − 3.74i)17-s + (−1.93 − 1.11i)19-s − 1.96i·23-s − 4.71·25-s + (−4.55 + 2.49i)27-s + (−5.16 − 2.98i)29-s + (0.800 + 0.462i)31-s + (6.37 + 2.37i)33-s + (−3.89 + 6.75i)37-s + ⋯ |
L(s) = 1 | + (0.349 − 0.937i)3-s − 0.238·5-s + (−0.756 − 0.654i)9-s + 1.18i·11-s + (−0.0324 + 0.0187i)13-s + (−0.0832 + 0.223i)15-s + (−0.524 − 0.908i)17-s + (−0.442 − 0.255i)19-s − 0.410i·23-s − 0.943·25-s + (−0.877 + 0.480i)27-s + (−0.959 − 0.553i)29-s + (0.143 + 0.0829i)31-s + (1.10 + 0.413i)33-s + (−0.641 + 1.11i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4366747476\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4366747476\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.604 + 1.62i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.533T + 5T^{2} \) |
| 11 | \( 1 - 3.92iT - 11T^{2} \) |
| 13 | \( 1 + (0.116 - 0.0674i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.93 + 1.11i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.96iT - 23T^{2} \) |
| 29 | \( 1 + (5.16 + 2.98i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.800 - 0.462i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.89 - 6.75i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.59 + 7.95i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.24 + 5.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.04 + 5.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.54 - 5.50i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.89 - 3.28i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.35 + 5.39i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.75 - 9.97i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.22iT - 71T^{2} \) |
| 73 | \( 1 + (-0.329 + 0.190i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.60 + 7.97i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.28 - 2.21i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.56 - 14.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.6 - 7.89i)T + (48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.789111714040410549511197190112, −7.996462230999748908669687366470, −7.17875029936071241187009933137, −6.79636711033426592872698707378, −5.72404092009593776790967135057, −4.71382515282911961646625754195, −3.73497041652716597120788559714, −2.53332040483923250275858204487, −1.77341486006727847717514540371, −0.14325553293528411795140695628,
1.86870171444045531589921975800, 3.19175377927727947953967531308, 3.78733675674407959344207574159, 4.69060973288601623310879540799, 5.69931722216149003527639307689, 6.31099104105273059626068026487, 7.65305557498034895487625746053, 8.271098051556959590306149687276, 8.936771325265259288954540333884, 9.663506001677088507603170684310